Question

which best describes the graph of the cubic function $f(x) = x^3 + x^2 + x + 1$?

• as $x$ increases, $y$ increases along the entire graph.
• as $x$ increases, $y$ increases, decreases, and then increases again.
• as $x$ increases, $y$ decreases, increases, and then decreases again.
• as $x$ increases, $y$ decreases along the entire graph.

Explanation:

Step1: Analyze the derivative

To determine the behavior of the cubic function \( f(x)=x^{3}+x^{2}+x + 1\), we first find its derivative. The derivative of \( f(x)\) using the power rule is \( f^{\prime}(x)=3x^{2}+2x + 1\).

Step2: Analyze the discriminant of the derivative

For a quadratic function \( ax^{2}+bx + c\), the discriminant is \( \Delta=b^{2}-4ac\). For \( f^{\prime}(x)=3x^{2}+2x + 1\), \( a = 3\), \( b=2\), and \( c = 1\). So, \( \Delta=(2)^{2}-4\times3\times1=4 - 12=- 8\). Since the discriminant \( \Delta<0\) and \( a = 3>0\), the quadratic function \( f^{\prime}(x)\) is always positive (because the parabola opens upwards and has no real roots, so it is above the x - axis for all real x).

Step3: Determine the function's behavior

If the derivative \( f^{\prime}(x)>0\) for all \( x\) in the domain of \( f(x)\) (which is all real numbers for a polynomial function), then the function \( f(x)\) is strictly increasing on its entire domain. That means as \( x\) increases, \( y=f(x)\) increases along the entire graph.

Answer:

As \( x\) increases, \( y\) increases along the entire graph. (The option corresponding to this description)